$Use Green’s theorem to evaluate the line integral $\int_C (1+xy^2)dx-x^2ydy$ on the arc of a parabola$

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$Use Green’s theorem to evaluate the line integral $\int_C (1+xy^2)dx-x^2ydy$ on the arc of a parabola$

Use Green’s theorem to evaluate the line integral
$$\int_C (1+xy^2)dx-x^2ydy,$$
where $C$ is the arc of the parabola $y = x^2$ from $(−1, 1)$ to $(1, 1)$.

Calculus Multivariable Calculus Integrals

Karl Sch

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Let $C'$ be the closed curve that is the union of C and the line segment from (1,1) to (-1,1). Then using the Green's theorem
\[\int_{C'}(1+xy^2)dx -x^2y dy =\iint_D -(2xy)+2xy dxdy=0.\]
Hence
\[\int_{C}(1+xy^2)dx -x^2y dy=-\int_{C''}(1+xy^2)dx -x^2y dy,\]
where $C''$ is the line segment from (1,1) to (-1,1) on which $y=1$. We can parametrize $C''$ as $(x,1)$ with $-1\leq x\leq 1$. So
\[-\int_{C''}(1+xy^2)dx -x^2y dy=\int_{-1}^{1}(1+x)dx=2.\]

Hence
\[\int_{C'}(1+xy^2)dx -x^2y dy =2.\]

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$Use Green’s theorem to evaluate the line integral $\int_C (1+xy^2)dx-x^2ydy$ on the arc of a parabola$

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